gsvd: Generalized Singular Value Decomposition

Description

Computes the generalized singular value decomposition of a pair of matrices.

Usage

gsvd(A,B)

Arguments

a matrix with $m$ rows and $n$ columns.

a matrix with $p$ rows and $n$ columns.

Value

The return value is a list containing the following components

A: the upper triangular matrix or a part of $R$.
B: lower part of the triangular matrix $R$ if $k+l>m$ (see below).
m: number of rows of $A$.
k: $r{-}l$. The number of rows of the matrix $R$ is $k{+}l$. The first $k$ generalized singular values are infinite.
l: effective rank of the input matrix $B$. The number of finite generalized singular values after the first $k$ infinite ones.
alpha: a numeric vector with length $n$ containing the numerators of the generalized singular values in the first $(k{+}l)$ entries.
beta: a numeric vector with length $n$ containing the denominators of the generalized singular value in the first $(k{+}l)$ entries.
U: the matrix $U$.
V: the matrix $V$.
Q: the matrix $Q$.

For a detailed description of these items see http://www.netlib.org/lapack/lug/node36.html. Auxiliary functions are provided for extraction and manipulation of the various items.

Details

The matrix A is a $m$-by-$n$ matrix and the matrix B is a $p$-by-$n$ matrix. This function decomposes both matrices; if either one is complex than the other matrix is coerced to be complex.

The Generalized Singular Value Decomposition of numeric matrices $A$ and $B$ is given as $$A = U D_1 [0\, R] Q^T$$ and $$B = V D_2 [0\, R] Q^T$$ where

$U$: an $m\times m$ orthogonal matrix.
$V$: a $p\times p$ orthogonal matrix.
$Q$: an $n\times n$ orthogonal matrix.
$R$: an $r$-by-$r$ upper triangular non singular matrix and the matrix $[0\, R]$ is an $r$-by-$n$ matrix. The quantity $r$ is the rank of the matrix $\left( \begin{array}{c} A \\B \end{array} \right)$ with $r \le n$.
$D_1$,$D_2$: are quasi diagonal matrices and nonnegative and satisfy $D_1^T D_1 + D_2^T D_2 = I$. $D_1$ is an $m$-by-$r$ matrix and $D_2$ is a $p$-by-$r$ matrix.

The Generalized Singular Value Decomposition of complex matrices $A$ and $B$ is given as $$A = U D_1 [0\, R] Q^H$$ and $$B = V D_2 [0\, R] Q^H$$ where

$U$: an $m\times m$ unitary matrix.
$V$: a $p\times p$ unitary matrix.
$Q$: an $n\times n$ unitary matrix.
$R$: an $r$-by-$r$ upper triangular non singular matrix and the matrix $[0\, R]$ is an $r$-by-$n$ matrix. The quantity $r$ is the rank of the matrix $\left( \begin{array}{c} A \\B \end{array} \right)$ with $r \le n$.
$D_1$,$D_2$: are quasi diagonal matrices and nonnegative and satisfy $D_1^T D_1 + D_2^T D_2 = I$. $D_1$ is an $m$-by-$r$ matrix and $D_2$ is a $p$-by-$r$ matrix.

For details on this decomposition and the structure of the matrices $D_1$ and $D_2$ see http://www.netlib.org/lapack/lug/node36.html.

References

Anderson. E. and ten others (1999) LAPACK Users' Guide. Third Edition. SIAM. Available on-line at http://www.netlib.org/lapack/lug/lapack_lug.html. See the section Generalized Eigenvalue and Singular Value Problems (http://www.netlib.org/lapack/lug/node33.html) and the section Generalized Singular Value Decomposition (GSVD) (http://www.netlib.org/lapack/lug/node36.html).

Examples

Run this code

# NOT RUN {
A <- matrix(c(1,2,3,3,2,1,4,5,6,7,8,8), nrow=2, byrow=TRUE)
B <- matrix(1:18,byrow=TRUE, ncol=6)
A
B

z <- gsvd(A,B)
z
# }

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